Multisection in the Stochastic Block Model Using Semidefinite Programming

Abstract

We consider the problem of identifying underlying community-like structures in graphs. Toward this end, we study the stochastic block model (SBM) on k-clusters: a random model on n = km vertices, partitioned in k equal sized clusters, with edges sampled independently across clusters with probability q and within clusters with probability p, p > q. The goal is to recover the initial “hidden” partition of [n]. We study semidefinite programming (SDP)-based algorithms in this context. In the regime $p = \frac {\alpha \log (m)}{m}$ and $q = \frac {\beta \log (m)}{m}$, we show that a certain natural SDP-based algorithm solves the problem of exact recovery in the k-community SBM, with high probability, whenever $\sqrt {\alpha } - \sqrt {\beta } > \sqrt {1}$, as long as $k=o(\log n)$. This threshold is known to be the information theoretically optimal. We also study the case when $k=\theta (\log (n))$. In this case however, we achieve recovery guarantees that no longer match the optimal condition $\sqrt {\alpha } - \sqrt {\beta } > \sqrt {1}$, thus leaving achieving optimality for this range an open question.

1 Appendix

Forms of Chernoff Bounds and Hoeffding Bounds Used in the Arguments

Theorem 7 (Chernoff)

Suppose X ₁…X _n be independent random variables taking values in {0, 1}. Let X denote their sum and let $\mu = \mathbb {E}[X]$ be its expectation. Then for any δ > 0 it holds that

$$\displaystyle \begin{aligned} \mathbb{P}\left( X > (1 + \delta)\mu\right) < \left(\frac{e^{\delta}}{(1 + \delta)^{(1+\delta)}}\right)^{\mu}\:, \end{aligned} $$

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$$\displaystyle \begin{aligned} \mathbb{P}\left( X < (1 - \delta)\mu\right) < \left(\frac{e^{-\delta}}{(1 - \delta)^{(1-\delta)}}\right)^{\mu} \:. \end{aligned} $$

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A simplified form of the above bound is the following formula (for δ ≤ 1)

$$\displaystyle \begin{aligned}\mathbb{P}\left( X \geq (1 + \delta)\mu\right) \leq e^{-\frac{\delta^2 \mu}{3}}\:, \end{aligned}$$

$$\displaystyle \begin{aligned}\mathbb{P}\left( X \leq (1 - \delta)\mu\right) \leq e^{-\frac{\delta^2 \mu}{2}} \:.\end{aligned}$$

Theorem 8 (Bernstein)

Suppose X ₁…X _n be independent random variables taking values in [−M, M]. Let X denote their sum and let $\mu = \mathbb {E}[X]$ be its expectation, then

$$\displaystyle \begin{aligned} \mathbb{P}\left( |X - \mu| \geq t \right) \leq \exp\left(-\frac{1}{2}\frac{t^2}{\sum_i \mathbb{E}[(X_i - \mathbb{E}[X_i])^2] + Mt/3}\right) \:. \end{aligned}$$

Corollary 1

Suppose X ₁…X _n are i.i.d Bernoulli variables with parameter p. Let σ = σ(X _i ) = p(1 − p); then we have that for any r ≥ 0

$$\displaystyle \begin{aligned}\mathbb{P}\left(X \geq \mu + \alpha\sigma\sqrt{n\log(r)}+ \alpha\log(r)\right) \leq e^{-\frac{\alpha\log(r)}{4}} \:.\end{aligned}$$

Proof

We have that nσ ² = np(1 − p) and M = 1. We can now choose $t = \alpha \sigma \sqrt {n\log (r)} + \alpha \log (r)$. This implies that $\frac {n\sigma ^2 + t/3}{t^2} \leq \frac {1}{\log (r)}\left (1/\alpha ^2 + 1/3\alpha \right ) \leq \frac {2}{\alpha \log (r)}$ which implies from Theorem 8 that $\mathbb {P}\left (X > \mu + \alpha \sigma \sqrt {n\log (r)}+ \alpha \log (r)\right ) \leq e^{-\frac {\alpha \log (r)}{4}}.$ □

Theorem 9 (Hoeffding)

Let X ₁…X _n be independent random variables. Assume that the X _i are bounded in the interval [a _i , b _i ]. Define the empirical mean of these variables as

$$\displaystyle \begin{aligned} \bar{X} = \frac{\sum_i \bar{X_i}}{n} \:, \end{aligned}$$

then

$$\displaystyle \begin{aligned} \mathbb{P}\left( |\bar{X} - \mathbb{E}[\bar{X}]| \geq t \right) \leq 2\exp\left(- \frac{2n^2t^2}{\sum_{i = 1}^{n} (b_i - a_i)^2}\right) \:. \end{aligned} $$

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